some note about the existence of cycles and chaotic solutions in economic models

MDEF 2008 BEATRICE VENTURI

1

SOME NOTE ABOUT THE EXISTENCE OF CYCLES AND

CHAOTIC SOLUTIONS IN ECONOMIC MODELS

by

Beatrice Venturi

Department of Economics

University of Cagliari

Italy


2

EXISTENCE OF CYCLE AND CAOTIC SOLUTIONS IN

ECONOMIC-FINANCIAL MODELS

We analyze the global structure of a tree-dimensional abstract continuous time stationary

economic model that includes some

determinates parameters.


3

We shall consider a generic

non-linear first order system with some structural parameters:

nii

xxxi

fx ...2,1,,,,,,,3211

3212

,,,, xxxgxi

3213

,,,, xxxhxi

THE MODEL


4

Where f, g and h are complicate non-linear functions of class C2 (twice continuously

differentiable) in all their arguments.

The parameters:

0, i nii ...2,1,

are real and positive.


5

A stationary (equilibrium) point of our system is any solution of :

0321 xxx

Assuming the existence of such

a solution at some point

*3

,*2

,*1

* xxxP


6

THE JACOBIAN MATRIX

The local dynamical properties of system from (2.1) to (2.3), at a hyperbolic

equilibrium point P*, can be described in terms of the Jacobian matrix , for brevity.

In fact, the nature of the eigenvalues of J*, plays a key role.

J P J* *


7

We consider the system as a one-parameter

family of differential equations dependent

of the parameter .

We fixe the other parameters .

We assume that in our modelexists a parameters set where the Jacobian

has two eigenvalues complex conjugate :

),(

J P J* *


8

EMERGENCE OF STABLE CYCLES

First we use rigorous arguments to show as anequilibrium of the model could be destabilizedinto a stable cycle in the dynamic of R3 under the

following two alternative assumptions:

a) A steady state has three stable roots.

b) A closed orbit has a two dimensional manifolds in which it is asymptotically stable .


9

APPLICATIONS

Next we apply these results to a general non-linear fixed-price disequilibrium

IS-LM model as formulated by

Neri U. and Venturi B. (2007).


10

APPLICATIONS

Neri U. and Venturi B. (2007) discuss the effect of a change of the adjustment parameter in the money market, via the Hopf bifurcation’s approach in a three dimensional fixed-price disequilibrium IS-LM model.


11

THE ECONOMIC MODEL

mwyrLr ,,.

S = savings, T= tax collections, G=government expenditure, B = interests payment on

perpetuities, L = liquidity preference

function m = is the real money

supply. I = investment, r =interest rate, y = output (income) w =wealth α >0 and µ>0are the adjustment parameters

in their respective markets

ByTGwySyrIy D ,,.

)(.

yTBGm

Specifically, our “generalized model” is a non-linear systemin the independent state variables r, y and m.

(1)

(2)

(3)


12

(1) describes the traditional disequilibrium of dynamic adjustment in the product market; (2) describes the corresponding disequilibrium in the money market;

(3) represents the governmental budget constraint.

THE ECONOMIC MODEL


13

THE ECONOMIC MODEL

Next, we define the disposable income y and the wealth w as follows:

ByTByyD

r

Bmw


14

THE ECONOMIC MODEL

We assume that the functions: I, S, T are of class C2

Recall that a stationary (equilibrium) point of our system is any solution of.

Assuming the existence of such a solution at some point P*(y*,r*,m*), we want to analyze its local properties (e.g. stability, etc.) around P*.

0 mry


15

myrfmwyrLr ,,,,

myrgByTGwyySyrIy D ,,,)(,

yhyTBGm )(

We shall rewrite our system in the equivalent form:

with h: R R

THE ECONOMIC MODEL


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Since G and B are fixed a choice of the policies, implies:

because all deficit must be financed either by creation

of money or by creation of new debt.

y

Tyh '

Remark1: Unlike Schinasi G.J. ,1982 , we allow the functions L and S (liquidity and savings) to depend on

wealth w.

THE ECONOMIC MODEL


17

THE ECONOMIC MODEL

The monotonicity, of the restrictions:

r I r ,.,. r L r ,.,. )(.,.,wLw ,.., ySy wSw .,.,

is assumed and these assumptions imply the (economic) conditions:

0rI

,

0rL

,

0wL

,

0yS 0wS

.


18

Dynamical analysis

Theorem 1. A hyperbolic stationary point of the system (1), (2), (3) is locally asymptotically stable if the following assumptions hold at:

Ngrf y 1

with N given by (*) at P* :

N = the first integer such that N a2> a3 , at P*,


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Dynamical analysis

yT

yTyS

yI

1 0yg

rww gLrfS )1( 10 wL gr 0

yT

wS

yL

rg

yg

rf 1

i) the marginal propensity to invest out of income is greater than unity

ii)

iii)

10 yT


20

Dynamical analysis

Corollary 1. Let the hypotheses of Theorem 1 hold at a hyperbolic stationary point of the our system.

Then the steady state for each α >0 and all

0**1

PNP

yg

rf

is locally asymptotically stable.


21

Dynamical analysis

Corollary 2. Assume Corollary 1 and let J*=J(P*) as before.

Then, there exists a value µ>0 for which

J* has a pair of purely imaginary eigenvalues.


22

Theorem 2. Assume the hypotheses of Theorem1 except that Ngrf y

1

is now replaced by :

*

*

20

Pr

f

Py

g

Then, there exists a continuous function µ(δ) with µ(0)=μ’ and for all δ small enough , there exists a continuous family of non-constant positive periodic solutions

[r*(t ,), y*(t,), m*(t,)]

for the dynamical system (1), (2), (3) which collapse to the stationary point P * .


23

REMARK For a three-dimensional non linear dynamical model the general version

of the Hopf bifurcation theorem, ensures the existence of a

small amplitude periodic solutions bifurcating

from the steady state only in the center manifold

(a two-dimensional subspace of 3).

THE ECONOMIC MODEL


24

Dynamical analysis

From an economic point of view, sub-critical or super-critical orbits are both reasonable.

Since the third real root of the Jacobian matrix is negative the existence of a super-critical Hopf bifurcations becomes very interesting in the analysis of macroeconomic fluctuations for a IS-LM model

A stable economy, by the increase or decrease of its control parameters, could be destabilized into a stable cycle in the dynamic of R3


25

CONCLUSIONS

The sub-critical Hopf bifurcations may correspond to the Keynesian corridor (Leijonhufvud, 1973):

the economy has stability inside the corridor while it will loose the stability outside the corridor.

In such a case the dynamics are either converging to an equilibrium point or the trajectories go somewhere else, and it is also possible that another attracting set exists, but often the alternative is diverging trajectories.


26

CONCLUSIONS

We have seen that fluctuations derive from the mechanisms through which money markets reflect and respond to the developments in the real economy.

Our analysis provides an example of the classical thesis concerning endogenous explanations to the existence of fluctuations in some real world economic variables


27

For a three-dimensional non linear dynamical model the general version of the Hopf

bifurcation theorem, ensures the existence of a small amplitude periodic solutions

bifurcating from the steady state only in the center manifold

(a two-dimensional subspace of 3).

The general version of the

Hopf bifurcation theorem


28

Shil’nikov showed that if the real eigenvalues has large magnitude than the real part of the

complex eigenvalues of the Jacobian of system,

then there are horseshoes present in return maps near the homoclinic

orbit of the model.

Shil’nikov Theorem


29

The orbits, in the dynamics of the center manifold, can generally be either

attracting or repelling..In the case of an attracting orbit

(the so- called sub-critical case) trajectory trajectory on the center manifoldcenter manifold are locallylocally

attracted by this this orbit, which becomes a limit set.

.


30

In this situation the stationary pointstationary point is an unstable solutionunstable solution

meanness from an economic point of viewmeanness from an economic point of view(unless the initial conditionsinitial conditions happen to coincidecoincide with the stationary value).

Conversely, if the cyclecycle is unstableunstable((the so- called super-critical case)

the steady state is attracting.


31

The study of stability of emerging orbits on the centre manifold can be performed

by calculating the sign up of-up-third order derivative of the nonlinear part of the system,

when written in normal form.

This case is particularly relevant from an economic point of view

only if the initial conditions imply that

the economy fluctuates right from the beginning.


32

In fact, the Hopf bifurcation theorem proves the existence of closed orbits but it gives no information on their number and

their stability.

Using the non linear parts of an equation system,

a stability coefficient

(as formulate for example by Guckenheimer J.- Holmes P.,1983)

may be calculated in order to determine

the stability properties of the closed orbits

(see Foley , 1989, Feichtinger ,1992, Mattana - Venturi,1999, Anedda C. -Venturi B. ,2003).


33

Mattana P.- Venturi B. , 1999, analyzed a simplified three-dimensional version of Lucas’s model,

a two sector endogenous growth model with externality.

Considering the externality as a bifurcation parameter, they proved, the existence of small

amplitude periodic solutions, Hopf bifurcating from the steady state in the center manifold.

Venturi B., 2002, have elaborated a numerical

simulation of this model.

.


34

In Fig.1 is plotted the dynamics of one orbit Hopf bifurcating from the steady state

of the reduced version of

the Lucas model

(see Venturi B., 2002)

The orbits is super-critical


35

Figure 1


36

THE ECONOMIC MODELS

We review a generalized two sector models of endogenous growth, with externalities, as formulated by Mulligan B.- Sala-I-Martin X.,1993.

We show that in this class of economic models, considering the externalityexternality as bifurcation parameterbifurcation parameter, the conditionsconditions of existenceexistence of periodic orbitsperiodic orbits and chaotic chaotic solutionssolutions come true.


37

The Mulligan B. - Sala-I-Martin X. model deal with the maximization of a standard utility function:

where

c = is per-capita consumption

= is a positive discount factor

= is the inverse of the intertemporal elasticity of substitution.

dtec t

0 1

1max

1


38

The constraints to the growth process are represented by the following equations

00

)0(,00

)0(

)(

)(ˆ)(ˆ))())(1)(()())(1((

)()(

)(ˆ)(ˆ))()()()()((

ˆˆ

ˆˆ

hhkk

thh

tkthtktvthtuBh

tctkk

tkthtktvthtuAk

khkuhu

khkvhu

k =is the physical capitalphysical capital,

h =is the human capitalhuman capital


39

vkuh ,

k h k h 1 1

k, , h are the private share of physicalphysical and the

humanhuman capital in the output sectoroutput sector

vkuh ,

k , h are the private shareprivate share of humanhuman

capitalcapital in the educationeducation sector sector


40

u and v =are the fraction of aggregate human human and physical capitalphysical capital

used in the final output sectorfinal output sector at instant t

(1- u) and (1- v)

are the fractionsfractions used in the education sector, education sector,

A and B

are the level of the technologylevel of the technology in each sectorsector,


41

is a positive externalityexternality, parameterparameter in the production of production of physical capital

k

h

is a positive externality parameterexternality parameter in the production of production of human capital

All the parameters:


42

live inside the following set:

= (0, 1)(0, 1) (0, 1) (0, 1) (0, 1) (0,1) (01) (0 1) (0 1) 4.


43

The representative agent’s problem is solved by defining the current value Hamiltonian:

)()(ˆ

)(ˆˆ

)(ˆ))()()()()((1

1

11

tctkk

ktkhthktkvtvhthutuA

cH

)(

ˆ)(ˆ

ˆ)(ˆ))())(1)(()())(1((

2th

hktkhthktkutvhthutuB

i, i = 1, 2 =Lagrange multipliers

(co-state variables). = is a depreciation factor


44

Plus the usual two transversality conditionstransversality conditions::

lim ( ) ( )

lim ( ) ( )

te t t k t

te t t h t

1 0

2 0

and obtaining the first-order necessary conditions for an interior solutionsinterior solutions.


45

The general model just presented collapses to Uzawa-Lucas’ when depreciation is neglected and the following

restrictions are imposed: v k

00ˆˆ kvhk

u h k 1

u h 1

A B 1


46

According to the strategy used by Mulligan B.C. - Sala-I-Martin X.,1993,we express the two multipliers 1 and 2

in terms of their corresponding control variables c and u

and obtain an autonomous system of four differential equations in the four variables k, h, c, u.

A solution of this autonomous system is called a Balanced Growth Path (BGP) if it entails a set of functions of time

solving the optimal control problem presented such that

k, h and c grow at a constant rate and u is a constant.


47

We choose a standard combination of the original variables that is stationary on BGP.

khx

1/1

1

ux 2

kcx

3

We get a first order autonomous system of

three differential equations


48

B.G.P that does not include positive externalities admits only the saddle-path stable solutions

(see Mulligan B.C. - Sala-I-Martin X.,1993).

We get a first order autonomous system of three differential equations

311)21(

12

111 xxxxxxx

322

1222 xxxxx

3312

11

233 xxxxxx


49

Where:

;

.1

1


50

We put our model into normal form(see Guckenheim - Holmes 1983, pp. 573).

The system becomes:

,13

,1

,111

3232

32322

32

,,*

,,*

,,*

3

2

1

wwwFwwBJw

wwwFwBJww

wwwFwTrJw


51

In the equilibrium point is translated the origin P*(0,0,0) of the model and the eigenvalues are

linearized in P*.

The real eigenvalue is

*TrJR

and the complex conjugate eigenvalues are

2,1 jBij


52

We consider the dynamic of the system when the parameter values are evaluated in a parameters

set where emerges a super-critical Hopf bifurcation.

We choose the value (the exponent of the externality factor )very close to the bifurcation value *.

Setting: = 0.75; = 0.055; = 0.054; = 0.1 and = 0.05,

We found that the system has a homoclinic orbit.

(See Fig. 3.)


53

Figure 2

00.05

0.10.15

0.20.25

-40

-20

0

20-35

-30

-25

-20

-15


54

The loss of stability and the presence of periodic solutions Hopf bifurcating from

the steady state can lead to the occurrence of an homoclinic orbit, Fig. 2,

and, under Shil’nikov assumption,

of chaotic solutions


55

The aim of this work is to point out some basic ideas that may be useful to prove the transition to

bounded and complex behavior, and to explain how

the presence of Hopf bifurcations in a general class of economic-financial models can be

interesting from an economic and dynamic point of view.

some note about the existence of cycles and chaotic solutions in economic models

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